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Paras31

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1 years, 93 days
Hellenic Open University
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Teacher of Mathematics with a proven track record of working in education management. Proficient in Ease of Adaptation, Course Design, and Instructional Technology. Holds a Bachelor's degree in Mathematics from the University of the Aegean and a Master's in Applied Mathematics at the Hellenic Open University, focusing on Ordinary and Partial Differential Equations. His enthusiasm lies in the application of mathematical models to real-world contexts, such as epidemiology and population growth. Aside from his passion for teaching, Athanasios enjoys football, basketball, and spending time with his dogs.

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Series of Complex Analysis Worksheets...

Paras31 245 Maple 2025
complex-analysis series convergence
April 28 2025
1 0

Hello everyone,

I am creating this post to begin a thread where I will share a series of worksheets on important topics in Complex Analysis, written as part of my notes for my classes. Complex_Analysis_Notes.pdf

The planned sections include:

  • Section 1: Infinite Series

  • Section 2: Power Series

  • Section 3: The Radius of Convergence of a Power Series

  • Section 4: The Riemann Zeta Function and the Riemann Hypothesis

  • Section 5: The Prime Number Theorem

Each worksheet will include calculations, plots, and examples using Maple to illustrate key ideas.

✅ I plan to upload one worksheet every week to keep a steady pace and allow time for discussion and feedback between posts.

I hope this thread will be helpful both for learning and for deeper exploration.
Feel free to comment, suggest improvements, or ask questions as I post the materials.

Thank you!

> 

restart; interface(imaginaryunit = 'I'); z := I*(1/3); S_N := proc (n) options operator, arrow; sum(z^k, k = 0 .. n) end proc; limit(S_N(n), n = infinity); S_N(10); S_N(100); S_N(1000); with(plots); points := [seq([Re(evalf(S_N(n))), Im(evalf(S_N(n)))], n = 0 .. 50)]; pointplot(points, connect = true, symbol = solidcircle, symbolsize = 10, color = blue, labels = ["Re", "Im"])

proc (n) options operator, arrow; sum(z^k, k = 0 .. n) end proc

  

9/10+(3/10)*I

  

53144/59049+(5905/19683)*I

  

 
> 

restart; interface(imaginaryunit = 'I'); z := I*(1/2); S_N := proc (n) options operator, arrow; sum(z^k, k = 0 .. n) end proc; limit(S_N(n), n = infinity); S_N(10); S_N(100); S_N(1000); with(plots); points := [seq([Re(evalf(S_N(n))), Im(evalf(S_N(n)))], n = 0 .. 50)]; pointplot(points, connect = true, symbol = solidcircle, symbolsize = 10, color = blue, labels = ["Re", "Im"])

proc (n) options operator, arrow; sum(z^k, k = 0 .. n) end proc

  

4/5+(2/5)*I

  

819/1024+(205/512)*I

  

 
> 

NULL
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; sum(((1/2)*I)^n, n = 0 .. N) end proc; fullsum := sum(((1/2)*I)^n, n = 0 .. infinity); realpts := [seq([n, Re(S(n))], n = 0 .. 30)]; imagpts := [seq([n, Im(S(n))], n = 0 .. 30)]; limit(Re(S(n)), n = infinity); limit(Im(S(n)), n = infinity); horiz_reallimit := plot(4/5, k = 0 .. 30, color = black, linestyle = 2, thickness = 2); horiz_imaglimit := plot(2/5, k = 0 .. 30, color = black, linestyle = 2, thickness = 2); plots[display]([pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Value"], legend = "Real Part"), pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Value"], legend = "Imaginary Part"), horiz_reallimit, horiz_imaglimit], axes = boxed, labels = ["n", "Partial Sum Value"])

4/5+(2/5)*I

  

4/5

  

2/5

  

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); H := proc (N) local n; sum(1/n, n = 1 .. N) end proc; limit(H(n), n = infinity); limit(Re(H(n)), n = infinity); limit(Im(H(n)), n = infinity); harmonic_pts := [seq([n, H(n)], n = 1 .. 100)]; harmonic_plot := pointplot(harmonic_pts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Partial Sum Value"], axes = boxed)

infinity

  

infinity

  

0

  

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; sum(I^k/k, k = 1 .. N) end proc; realpts := [seq([n, Re(S(n))], n = 1 .. 100)]; imagpts := [seq([n, Im(S(n))], n = 1 .. 100)]; complex_pts := [seq([Re(S(n)), Im(S(n))], n = 1 .. 100)]; S_infinite := sum(I^k/k, k = 1 .. infinity); Re(S_infinite); Im(S_infinite); horiz_reallimit := plot(-(1/2)*ln(2), k = 0 .. 100, color = black, linestyle = 2, thickness = 2); horiz_imaglimit := plot((1/4)*Pi, k = 0 .. 100, color = black, linestyle = 2, thickness = 2); real_plot := pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Partial Sum Value"], axes = boxed, legend = "Real Part"); imag_plot := pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Partial Sum Value"], axes = boxed, legend = "Imaginary Part"); plots[display]([real_plot, horiz_reallimit, imag_plot, horiz_imaglimit]); plots[pointplot](complex_pts, symbol = solidcircle, style = pointline, color = blue, axes = boxed, labels = ["Re", "Im"])

-(1/2)*ln(2)+((1/4)*I)*Pi

  

-(1/2)*ln(2)

  

(1/4)*Pi

  

 

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; sum((-(2/3)*I)^n, n = 0 .. N) end proc; fullsum := sum((-2*I*(1/3))^n, n = 0 .. infinity); realpts := [seq([n, Re(S(n))], n = 0 .. 30)]; imagpts := [seq([n, Im(S(n))], n = 0 .. 30)]; limit(Re(S(n)), n = infinity); limit(Im(S(n)), n = infinity); horiz_reallimit := plot(9/13, k = 0 .. 30, color = black, linestyle = 2, thickness = 2); horiz_imaglimit := plot(-6/13, k = 0 .. 30, color = black, linestyle = 2, thickness = 2); plots[display]([pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Value"], legend = "Real Part"), pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Value"], legend = "Imaginary Part"), horiz_reallimit, horiz_imaglimit], axes = boxed, labels = ["n", "Partial Sum Value"])

9/13-(6/13)*I

  

9/13

  

-6/13

  

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; sum((I*Pi)^n, n = 0 .. N) end proc; realpts := [seq([n, Re(S(n))], n = 0 .. 100)]; imagpts := [seq([n, Im(S(n))], n = 0 .. 100)]; complex_pts := [seq([Re(S(n)), Im(S(n))], n = 0 .. 100)]; limit(S(N), N = infinity); limit(Re(S(n)), n = infinity); limit(Im(S(n)), n = infinity); real_plot := pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Partial Sum (Real Part)"], title = "Real Part of Partial Sums of (Pi i)^n", axes = boxed); imag_plot := pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Partial Sum (Imaginary Part)"], title = "Imaginary Part of Partial Sums of (Pi i)^n", axes = boxed); complex_plot := pointplot(complex_pts, symbol = solidcircle, style = pointline, color = blue, labels = ["Re", "Im"], title = "Partial Sums in Complex Plane (Pi i)^n", axes = boxed)

undefined

  

undefined

  

undefined

  

 

 

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; sum(2*I^k/k, k = 1 .. N) end proc; realpts := [seq([n, Re(S(n))], n = 1 .. 100)]; imagpts := [seq([n, Im(S(n))], n = 1 .. 100)]; complex_pts := [seq([Re(S(n)), Im(S(n))], n = 1 .. 100)]; S_infinite := sum(2*I^k/k, k = 1 .. infinity); Re(S_infinite); Im(S_infinite); horiz_reallimit := plot(-ln(2), k = 0 .. 100, color = black, linestyle = 2, thickness = 2); horiz_imaglimit := plot((1/2)*Pi, k = 0 .. 100, color = black, linestyle = 2, thickness = 2); real_plot := pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Partial Sum Value"], axes = boxed, legend = "Real Part"); imag_plot := pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Partial Sum Value"], axes = boxed, legend = "Imaginary Part"); plots[display]([real_plot, horiz_reallimit, imag_plot, horiz_imaglimit]); plots[pointplot](complex_pts, symbol = solidcircle, style = pointline, color = blue, axes = boxed, labels = ["Re", "Im"])

-ln(2)+((1/2)*I)*Pi

  

-ln(2)

  

(1/2)*Pi

  

 

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; add(exp(Pi*I*n)/n, n = 1 .. N) end proc; realpts := [seq([n, Re(S(n))], n = 1 .. 100)]; imagpts := [seq([n, Im(S(n))], n = 1 .. 100)]; complex_pts := [seq([Re(S(n)), Im(S(n))], n = 1 .. 100)]; S_infinite := sum(exp(Pi*I*n)/n, n = 1 .. infinity); limit_Re := Re(S_infinite); limit_Im := Im(S_infinite); limit_Re; limit_Im; real_plot := pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Partial Sum Value"], title = "Real Part of Partial Sums", axes = boxed); imag_plot := pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Partial Sum Value"], title = "Imaginary Part of Partial Sums", axes = boxed); complex_plot := pointplot(complex_pts, symbol = solidcircle, style = pointline, color = blue, labels = ["Re", "Im"], title = "Partial Sums in Complex Plane", axes = boxed); plots[display]([real_plot, imag_plot]); plots[display](complex_plot)

-ln(2)

  

-ln(2)

  

0

  

-ln(2)

  

0

  

 

 
> 

restart; with(plots); interface(imaginaryunit = 'I'); S := proc (N) local n; add(exp(2*Pi*I*n), n = 0 .. N) end proc; realpts := [seq([n, Re(S(n))], n = 0 .. 100)]; imagpts := [seq([n, Im(S(n))], n = 0 .. 100)]; complex_pts := [seq([Re(S(n)), Im(S(n))], n = 0 .. 100)]; S_infinite := sum(exp(2*Pi*I*n), n = 1 .. infinity); limit_Re := Re(S_infinite); limit_Im := Im(S_infinite); real_plot := pointplot(realpts, symbol = solidcircle, style = pointline, color = blue, labels = ["n", "Partial Sum Value"], title = "Real Part of Partial Sums", axes = boxed); imag_plot := pointplot(imagpts, symbol = solidbox, style = pointline, color = red, labels = ["n", "Partial Sum Value"], title = "Imaginary Part of Partial Sums", axes = boxed); complex_plot := pointplot(complex_pts, symbol = solidcircle, style = pointline, color = blue, labels = ["Re", "Im"], title = "Partial Sums in Complex Plane", axes = boxed); plots[display]([real_plot, imag_plot]); plots[display](complex_plot)

infinity

  

infinity

  

0

  

 

 
 

``

Download infinite_series.mw

An Introduction to Limits with Maple...

Paras31 245 Maple 2024
calculus limit
March 21 2025
2 0

LIMITS

Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas indefinite integrals are expressed without limits, and it will have an arbitrary constant while integrating the function.

Sometimes we can't work something out directly ... but we can see what it should be as we get closer and closer!

Example 1

> 

"restart; f(x):=(|x|-3)/(x-3);"

proc (x) options operator, arrow, function_assign; (abs(x)-3)/(x-3) end proc

 (1)
> 

plot(f(x), x = -10 .. 10, discont = true, color = "Green")

 
> 

f(3)

Error, (in f) numeric exception: division by zero

  

Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.

So instead of trying to work it out for x=3 let's try approaching it closer and closer:

> 

f(3.01)

1.000000000

 (2)
> 

f(3.0000001)

1.000000000

 (3)
> 

f(2.9999999)

1.000000000

 (4)
> 

Limit(f(x), x = 3)

Limit((abs(x)-3)/(x-3), x = 3)

 (5)
> 

limit(f(x), x = 3)

1

 (6)
> 

limit(f(x), x = 3, left)

1

 (7)
> 

limit(f(x), x = 3, right)

1

 (8)

Example 2

Sometimes some functions are not continuous. That is, they appear to be approaching two different values when they are approached from two sides.

> 

"g(x):=piecewise(0<x<2,1/(2 x-x^(2)),2 <x<=3,2 -x,3<x<4,x-4, 4<=x,Pi,undefined);"

proc (x) options operator, arrow, function_assign; piecewise(0 < x and x < 2, 1/(2*x-x^2), 2 < x and x <= 3, 2-x, 3 < x and x < 4, x-4, 4 <= x, Pi, undefined) end proc

 (9)
> 

plot(g(x), x = -10 .. 10, y = -1 .. 10, discont = true, color = "Red")

 

Suppose we want to approach 2 and see the function’s limit. This naturally leads to directions from which we can approach. Left-hand side and the right-hand side limits.

The right-hand side limit is the value of the function that it takes while approaching it from the right-hand side of the desired point. Similarly, the left-hand side limit is the value of function while approaching it from the left-hand side.

> 

eval(g(x), x = 2)

undefined

 (10)
> 

limit(g(x), x = 2, left)

infinity

 (11)
> 

limit(g(x), x = 2, right)

0

 (12)
> 

limit(g(x), x = 2)

undefined

 (13)

And the ordinary limit "does not exist".

> 

g(4)

Pi

 (14)
> 

limit(g(x), x = 4, left)

0

 (15)
> 

limit(g(x), x = 4, right)

Pi

 (16)
> 

limit(g(x), x = 4)

undefined

 (17)

And the ordinary limit "does not exist".

> 

with(Student[Calculus1]); LimitTutor()

Example 3

Estimate the value of the following limit limit(h(x)*where, x = 2), h(x) = piecewise(x <> 2, x+12, x = 2, 4).

> 

"h(x):={[[x+12,x<>2],[4,x=2]];"

proc (x) options operator, arrow, function_assign; piecewise(x <> 2, x+12, x = 2, 4) end proc

 (18)
> 

plot(h(x), x = -10 .. 10, discont = true, color = "#40e0d0")

 
> 

limit(h(x), x = 2)

14

 (19)

The limit is NOT 2025!Remember from the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point. Since the only thing about the function that we actually changed was its behavior at x = 2 this will not change the limit.

Example 4

> 

" w(x):=piecewise( x<0,-x+5,x>=0,2 x);"

proc (x) options operator, arrow, function_assign; piecewise(x < 0, -x+5, 0 <= x, 2*x) end proc

 (20)
> 

plot(w(x), x = -10 .. 10, y = -10 .. 10, discont = true, color = "Blue")

 
> 

limit(w(x), x = 5)

10

 (21)
> 

limit(w(x), x = 6, left)

12

 (22)
> 

limit(w(x), x = 1, right)

2

 (23)

Example 5

> 

" k(x):=piecewise( x<5,x+4,x>=5, x^(2)-2);"

proc (x) options operator, arrow, function_assign; piecewise(x < 5, x+4, 5 <= x, x^2-2) end proc

 (24)
> 

plot(k(x), x = -10 .. 10, discont = true, color = orange)

 
> 

limit(k(x), x = 2)

6

 (25)
> 

limit(k(x), x = 5, left)

9

 (26)
> 

limit(k(x), x = 5, right)

23

 (27)
> 

limit(k(x), x = 5)

undefined

 (28)
> 

limit(k(x), x = 6)

34

 (29)

Example 6

> 

restart
> 

" l(x):=piecewise( x<=1,(x-8)/(x-3),x>=3, sqrt(x^(2)+x+2), undefined);"

proc (x) options operator, arrow, function_assign; piecewise(x <= 1, (x-8)/(x-3), 3 <= x, sqrt(x^2+x+2), undefined) end proc

 (30)
> 

plot(l(x), x = -10 .. 10, discont = true, color = "Blue")

 
> 

limit(l(x), x = 0)

8/3

 (31)
> 

limit(l(x), x = 1, left)

7/2

 (32)
> 

limit(l(x), x = 1, right)

undefined

 (33)
> 

limit(l(x), x = 2)

undefined

 (34)

Example 7

Estimate the value of the following limit. limit(H(t), t = 0)where, H(t) = piecewise(t < 0, 0, t >= 0, 1)

> 

" H(t):=piecewise( t<0,0,t>=0, 1);"

proc (t) options operator, arrow, function_assign; piecewise(t < 0, 0, 0 <= t, 1) end proc

 (35)

This function is often called either the Heaviside or step function. We could use a table of values to estimate the limit, but it’s probably just as quick in this case to use the graph so let’s do that. Below is the graph of this function.

> 

plot(H(t), t = -10 .. 10, discont = true, color = "Blue")

 
> 

limit(H(t), t = 0, left)

0

 (36)
> 

limit(H(t), t = 0, right)

1

 (37)

We can see from the graph that if we approach t = 0from the right side the function is moving in towards a yvalue of 1. Well actually it’s just staying at 1, but in the terminology that we’ve been using in this section it’s moving in towards 1.

Also, if we move in towards t = 0 from the left the function is moving in towards a yvalue of 0.

According to our definition of the limit the function needs to move in towards a single value as we move in towards t = a (from both sides). This isn’t happening in this case and so in this example we will also say that the limit doesn’t exist.

 

NULL

Download limits.mw

ΕΜΒΑΔΟΝ ΕΠΙΠΕΔΟΥ ΧΩΡΙΟΥ – Maple Workshee...

Paras31 245 Maple 2024
March 10 2025
1 2

Hello everyone,

I have created a Maple worksheet titled "ΕΜΒΑΔΟΝ ΕΠΙΠΕΔΟΥ ΧΩΡΙΟΥ", designed to help my students prepare for their final exams as they qualify for university. This worksheet focuses on area calculations in plane geometry, using Maple to visualize and solve problems efficiently.

This worksheet is aimed at high school students preparing for university entrance exams, as well as teachers who want to integrate Maple into their teaching.

I would love to hear your thoughts and feedback!

Have you used Maple for similar exam preparation?
εμβαδόν_χωρίου.mw

Simulating the Dynamics of a Novel Disea...

Paras31 245 Maple 2024
ode dsolve numeric
December 30 2024
0 0

In this activity, we are trying to simulate an outbreak of a new infectious disease that our population of 10^6people has not been exposed to before. This means that we are starting with a single case, everyone else is susceptible to the disease, and no one is yet immune or recovered. This can for example reflect a situation where an infected person introduces a new disease into a geographically isolated population, like on an island, or even when an infections "spill over" from other animals into a human population. In terms of the initial conditions for our model, we can define: "S=10^(6) -1=999999," I = 0and R = 0. NULL

Remember, the differential equations for the simple SIR model look like this:

dS/dt = `&lambda;S`*dI/dt and `&lambda;S`*dI/dt = `&lambda;S`-I*gamma, dR/dt = I*gamma

Initial number of people in each compartment
S = 10^6-1",I=0 "and R = 0.

> 

NULL

Parameters:

gamma = .1*recovery*rate*beta and .1*recovery*rate*beta = .4*the*daily*infection*rate

> 

restart; with(plots); _local(gamma)
> 

sys := diff(s(t), t) = -lambda*s(t), diff(i(t), t) = lambda*s(t)-gamma*i(t), diff(r(t), t) = gamma*i(t)

diff(s(t), t) = -lambda*s(t), diff(i(t), t) = lambda*s(t)-gamma*i(t), diff(r(t), t) = gamma*i(t)

 (1)
> 

ic := s(0) = s__0, i(0) = i__0, r(0) = r__0
> 

gamma := .1; beta := .4; n := 10^6

.1

  

.4

  

1000000

 (2)
> 

lambda := beta*i(t)/n
> 

s__0, i__0, r__0 := 10^6-1, 1, 0
> 

NULL
> 

sols := dsolve({ic, sys}, numeric, output = listprocedure)
> 

display([odeplot(sols, [t, s(t)], 0 .. 100, color = red), odeplot(sols, [t, i(t)], 0 .. 100, color = blue), odeplot(sols, [t, r(t)], 0 .. 100, color = green)], labels = ["Time [day]", "Population"], labeldirections = [horizontal, vertical], legend = ["Susceptible", "Infected", "Recovered"], legendstyle = [location = right])

 

Remember that in a simple homogenous SIR model, `R__eff `is directly related to the proportion of the population that is susceptible:

R__eff = R__0*S/N

> 

Reff := proc (t) options operator, arrow; beta*s(t)/(gamma*n) end proc
> 

odeplot(sols, [[t, Reff(t)]], t = 0 .. 100, size = [500, 300], labels = ["Time [day]", "Reff"], labeldirections = [horizontal, vertical])

 

The effective reproduction number is highest when everyone is susceptible: at the beginning, `R__eff ` = R__0. At this point in our example, every infected cases causes an average of 4 secondary infections. Over the course of the epidemic, `R__eff ` declines in proportion to susceptibility.

The peak of the epidemic happens when `R__eff ` goes down to 1 (in the example here, after 50 days). As `R__eff `decreases further below 1, the epidemic prevalence goes into decline. This is exactly what you would expect, given your understanding of the meaning of `R__eff ` once the epidemic reaches the point where every infected case cannot cause at least one more infected case (that is, when `R__eff ` < 1), the epidemic cannot sustain itself and comes to an end.

> 

susceptible := eval(s(t), sols); infected := eval(i(t), sols); recovered := eval(r(t), sols)
> 

susceptible(51)

HFloat(219673.04834159758)

 (3)
> 

infected(51)

HFloat(401423.4112878752)

 (4)
> 

recovered(51)

HFloat(378903.54037052736)

 (5)
> 

Reffe := proc (t) options operator, arrow; beta*susceptible(t)/(gamma*n) end proc

proc (t) options operator, arrow; beta*susceptible(t)/(gamma*n) end proc

 (6)
> 

Reffe(51)

HFloat(0.8786921933663903)

 (7)

Prevalence is simply the value of Iat a given point in time. Now we can see that the incidence is the number of new cases arriving in the I compartment in a given interval of time. The way we represent this mathematically is by taking the integral of new cases over a given duration.

For example, if we wanted to calculate the incidence from day 7 to 14,

int(`&lambda;S`(t), t = 7 .. 14)

> 

lamda := proc (t) options operator, arrow; beta*infected(t)/n end proc

proc (t) options operator, arrow; beta*infected(t)/n end proc

 (8)
> 

inflow := proc (t) options operator, arrow; lamda(t)*susceptible(t) end proc

proc (t) options operator, arrow; lamda(t)*susceptible(t) end proc

 (9)
> 

int(inflow(t), t = 7 .. 14)

HFloat(78.01804723222038)

 (10)
> 

incidence_plot := plot(inflow(t), t = 0 .. 14, color = orange, labels = ["Time (days)", "Incidence Rate"], labeldirections = [horizontal, vertical], title = "Incidence Rate between t=7 and t=14")

 
> 

s, i, r := eval(s(t), sols), eval(i(t), sols), eval(r(t), sols); T := 100; dataArr := Array(-1 .. T, 1 .. 4); dataArr[-1, () .. ()] := `<,>`("Day", "Susceptible", "Infected", "Recovered")


Assign all the subsequent rows

> 

for t from 0 to T do dataArr[t, () .. ()] := `~`[round](`<,>`(t, s(t), i(t), r(t))) end do

 

Tabulate through the DocumentTools

> 

DocumentTools:-Tabulate(dataArr, alignment = left, width = 50, fillcolor = (proc (A, n, m) options operator, arrow; ifelse(n = 1, "DeepSkyBlue", "LightBlue") end proc))

Download dynamics_of_novel_disease_outbreak.mw

Sample Worksheet on ODEs with Maple – A ...

Paras31 245 Maple 2024
ode differential-equations initial-conditions
November 19 2024
1 4

Hello Maple enthusiasts,

I am excited to share a sample worksheet on Ordinary Differential Equations (ODEs), created as part of my ongoing project—a book I am writing for undergraduate students. This book is designed to teach ODEs using Maple, offering an interactive and intuitive approach to solving differential equations.

As far as I know, there aren’t many books available in the Greek language that combine ODEs with Maple. In fact, I believe there’s only one other such resource, which highlights the lack of materials in this niche. My goal is to fill this gap by providing students and educators with a resource that is both practical and accessible, leveraging Maple's powerful capabilities to deepen understanding and simplify complex concepts.

The worksheet I’m sharing includes:

  • Step-by-step solutions to ODEs using Maple.
  • Graphical representations to visualize solutions, which I believe are invaluable for fostering comprehension.

I hope this preview sparks your interest and provides insight into the teaching style and structure of the upcoming book. I would love to hear your thoughts, feedback, or suggestions for topics you think should be included.

Solve the following differential equation

diff(y(x), x) = x*y(x)
with initial condition y(0) = 1

> 

restart; with(plots); with(DEtools)
> 

ode := diff(y(x), x) = x*y(x)

diff(y(x), x) = x*y(x)

 (1)
> 

ic := y(0) = 1

y(0) = 1

 (2)
> 

general_solution := dsolve(ode, y(x))

y(x) = c__1*exp((1/2)*x^2)

 (3)
> 

particular_solution := dsolve({ic, ode}, y(x))

y(x) = exp((1/2)*x^2)

 (4)
> 

soln := dsolve({ic, ode}, y(x), numeric)

``

 (5)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode, [y(x)], x = -5 .. 5, y = -5 .. 5, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -5 .. 5])

 

NULL

Solve the following differential equation

diff(y(x), x) = x/y(x)
with initial condition y(0) = 2

> 

restart; with(plots); with(DEtools)
> 

ode1 := diff(y(x), x) = x/y(x)

diff(y(x), x) = x/y(x)

 (6)
> 

ic := y(0) = 2

y(0) = 2

 (7)
> 

dsolve(ode1, y(x))

y(x) = (x^2+c__1)^(1/2), y(x) = -(x^2+c__1)^(1/2)

 (8)
> 

NULL
> 

soln := dsolve({ic, ode1}, y(x), numeric)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode1, [y(x)], x = -5 .. 5, y = -5 .. 5, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -5 .. 5])

 

NULL

Solve the following differential equation

dy/dx = e^y/(x^2+1)
with initial condition y(0) = -1

> 

restart; with(plots); with(DEtools)
> 

ode2 := diff(y(x), x) = exp(y(x))/(x^2+1)

diff(y(x), x) = exp(y(x))/(x^2+1)

 (9)
> 

ic := y(0) = -1

y(0) = -1

 (10)
> 

dsolve(ode2, y(x))

y(x) = ln(-1/(arctan(x)+c__1))

 (11)
> 

soln := dsolve({ic, ode2}, y(x), numeric)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode2, [y(x)], x = -5 .. 5, y = -5 .. 5, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -5 .. 5])

 

NULL

Solve the following differential equation

dy/dx = y^2+y
with initial condition y(1) = 2

> 

restart; with(plots); with(DEtools)
> 

ode3 := diff(y(x), x) = y(x)+y(x)^2

diff(y(x), x) = y(x)+y(x)^2

 (12)
> 

ic := y(1) = 2

y(1) = 2

 (13)
> 

dsolve(ode3, y(x))

y(x) = 1/(-1+exp(-x)*c__1)

 (14)
> 

soln := dsolve({ic, ode3}, y(x))

y(x) = 2/(-2+3*exp(-x)*exp(1))

 (15)
> 

DEplot(ode3, y(x), x = -5 .. 5, y = -5 .. 5, [[y(1) = 2]], color = blue, arrows = slim, scaling = constrained, axes = boxed)

 

NULL

Solve the following differential equation

y*dy/dx-x = 0
with initial condition y(0) = 4, y(1) = 2, y(-1) = -2and y(-2) = -4.

> 

restart; with(plots); with(DEtools)
> 

ode4 := y(x)*(diff(y(x), x))-x = 0

y(x)*(diff(y(x), x))-x = 0

 (16)
> 

ic := y(0) = 4

y(0) = 4

 (17)
> 

dsolve(ode4, y(x))

y(x) = (x^2+c__1)^(1/2), y(x) = -(x^2+c__1)^(1/2)

 (18)
> 

DEplot(ode4, y(x), x = -5 .. 5, y = -5 .. 5, [[y(1) = 2], [y(0) = 4], [y(-1) = -2], [y(-2) = -4]], color = blue, arrows = slim, scaling = constrained, axes = boxed)

 

 

 

NULL

Solve the following differential equation

dy/dx = 2*x+2*xy^2/y
with initial condition y(0) = 2.

> 

restart; with(plots); with(DEtools)
> 

ode5 := diff(y(x), x) = (2*x+2*x*y(x)^2)/y(x)

diff(y(x), x) = (2*x+2*x*y(x)^2)/y(x)

 (19)
> 

ic := y(0) = 2

y(0) = 2

 (20)
> 

dsolve(ode5, y(x))

y(x) = (exp(2*x^2)*c__1-1)^(1/2), y(x) = -(exp(2*x^2)*c__1-1)^(1/2)

 (21)
> 

psoln := dsolve({ic, ode5}, y(x))

y(x) = (5*exp(2*x^2)-1)^(1/2)

 (22)
> 

soln := dsolve({ic, ode5}, y(x), numeric)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode5, [y(x)], x = -5 .. 5, y = -3 .. 10, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -3 .. 10])

 
> 

NULL


 

Download separable_diff_equations.mw

 

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